A R K I V F O R M A T E M A T I K B a n d 3 n r 34 Communicated 9 February 1955 by O. H. FAX~I~
On measurement of velocity by Pitot tube By
BENGT J. ANDERSSONWith 4 figures in the text
1. We know several p a r a d o x a l results from theoretical h y d r o d y n a m i c s which are incompatible with practical experience. Obviously such a p a r a d o x can be explained b y an u n p e r m i t t e d simphfication in the theorizer's assumptions. I f e.g.
all b o u n d a r y conditions in a problem are s t a t i o n a r y or the b o u n d a r y has some s y m m e t r y properties, it is plausible to assume t h e same character for the solu- tion of the h y d r o d y n a m i c a l equations. Sometimes it is a good a p p r o x i m a t i o n to neglect the viscosity, e.g. b y calculation of the velocity outside an airfoil in a homogeneous flow, b u t the supposition is an over-simplification if we seek the resistance of the b o d y (d'Alembert paradox). F o r further examples of this kind I refer to a famous book b y GAR~TT BnU~_UOrF [1]. H e r e I shall discuss a problem where the effects of viscosity are problematical.
A real fluid is viscous, and this fact m a y cause accumulation of fluid in
" w a k e s " . I n some cases a wake m a y h a v e a r a t h e r well defined b o u n d a r y zone which on idealization to non-viscous fluid tends to a surface, called a " f r e e "
boundary, where the velocity is discontinuous. Sometimes t h e wake is bounded b y a t u r b u l e n t " m i x i n g zone", and the turbulence produces motions in the wake, which are practically inaccessible for theoretical analysis. E v i d e n t l y it is dif- ficult to predict the existence of wakes, and without a condition of s t a b i h t y , we get an infinite n u m b e r of solutions to a given problem. On account of the difficulty of surveying the s t a b i h t y problem for all conceivable cases, we m u s t in general supplement the theoretical speculations with experimental experiences.
2. A long, straight, circular t u b e with thin walls is closed b y a wall inside the tube. T h e t u b e is immersed in an incompressible fluid, and at a great distance from the end of the t u b e the flow is homogeneous, s t a t i o n a r y a n d parallel to the t u b e axis. The velocity is so great t h a t we m a y , as a first approximation, neglect the viscosity.
I f no wakes were accumulated the calculation of the flow (Fig. 1) should be a classical problem for harmonic functions. I n r e a h t y we m a y expect t h a t fluid is accumulated in the t u b e a n d eventually forms a stable wake before t h e wall.
P e r h a p s it is plausible t, h a t the free b o u n d a r y has rotational s y m m e t r y a b o u t the t u b e axis and forms a peak on the axis (Fig. 2). We shall not t r y to m a k e a stability analysis of this flow, b u t it is obvious t h a t internal friction will b r e a k down the p e a k a t an a r b i t r a r y small deviation from the s y m m e t r i c ar- rangement.
29 391
B. ~WDERSSON, On measurement of velocity by Pitot tube
~ / ~ _ _ -
~_ ~ - ~4/~ C*"
~ - ~ - - : = - - ~ [ , --~__~ - _ _ - -
Fig.. 1. Fig. 2.
I n order to establish the real behaviour of this flow the following e x p e r i m e n t was carried out. The tube, m a d e of glass a n d filled with coloured water, was immersed in a t a n k with flowing w a t e r a n d suitably arranged. A s t a t i o n a r y s t a t e was quickly a t t a i n e d a n d it was found t h a t t h e wake was n o t s y m m e t r i c a b o u t t h e t u b e axis b u t a p p a r e n t l y s y m m e t r i c a b o u t a plane t h r o u g h the axis (Fig. 3). Small disturbances of the direction of the t u b e caused a rotation of the s y m m e t r y plane round t h e axis.
I t seems to me t h a t the formation of the wake continues until the free b o u n d a r y reaches the front edge of the tube. I f the free b o u n d a r y extended b e y o n d the opening of the t u b e we should obtain a p e a k like t h a t in Fig. 2 a n d this p e a k m i g h t be broken down b y the turbulence t h a t always exists in the surrounding flow.
3. We s t u d y the two-dimensional ease, obtained b y substituting for the t u b e two thin, parallel plates of infinite extension (Fig. 4). At a great distance from t h e edges t h e flow is parallel to the plates. We assume t h a t the formation of a stable two-dimensional wake is completed and the free b o u n d a r y reaches the edge of one plate. We introduce curtesian coordinates x, y according to Fig. 4.
Let u = u 2 + v ~ be the velocity vector a n d p u t z = x + i y , T = u - i v . I f ¢(z) is the real velocity potential, ~ = grad ~0, a n d ~p (z) is the conjugate harmonic function to ~, t h e " s t r e a m function", t h e n W = ~ + i y~ is an analytic function, regular in the flow region Az outside the wake. F u r t h e r we know t h a t z = W' (z).
We can assume t h a t ~ = 0 on the free b o u n d a r y a n d 3 = 1 a t z = - o o . Ac- cording to Bernoulli's t h e o r e m the velocity is constant on t h e free boundary.
I n t h e W-plane, the region Az is represented "schlicht" on a region Aw, which is t h e W-plane cut along the positive real axis, and in the z-plane
"schlieht" on the region
Izl>k}
392
Fig. 3.
A R K I V FOIl MATEMATIK. Bd 3 nr 34
~
Y1 !
j Z-plane
c
~-plane
D
D'
t
A[
i
!
I
D' D
F i g . 4.
B ? C D
0
W-plane
where k is the constant velocity on the free boundary.
spondence of the boundaries is clear from Fig. 4.
The mapping A~--+Aw is given by
W (~) = K (3 - k) ~
(~" - - 1) 2 ( 3 - - k ~ ) 2
The detailed eorre-
where K is a positive constant. Hence follows
f f (T--k) 4 T - k + 1 - k ~ - k
z ( v ) = ~ - ' W ' ( ~ ) d 3 . = K I v @ _ l ) ~ ( ~ _ k ~ ) 2 + ~ ( f ~ 1 _ ~ +
/c
1 - k ~ - k k (2+k) 1 - ~ ~ T
+ k ( 1 + k ) ~ 3 _ k ~ + 2 ( 1 ~ (1 - T) l°g ~ _ k + 2 l ° g ~ c - - 2 ( l - k ) (1 + 2 k ) "t" - - k2 /
k ~ ( l + k ) 3 l o g k ( l _ k ) j .
(1)
At the edge C we have T = co and z = zc = i H, where H is the distance between the plates. We obtain from (1)
2
k (1 + k 2) k (1 - k) (2 + k)
Zc = K (1 + k) 2 - 2 (1 + k) 3 1 2 ( 1 - k ) 2 1 log I t - ~ l ° g 1 - ~ +
k (1 - k) (2 + k) /
j.
From R z c = 0 follows k=0.3270 and hence H = 1.377 K. The free boundary is given b y z = z ( k e ~ ) , O>_fl>_~r. Especially if f l = - ~ we get the point zp where the distance from the opening to a point on the free boundary has a maximum, and we get zp = (0.896 + 0.713. i) H.
393
B. ANDERSSON, 0/t measurement of velocity by Pitot tube
4. A c c o r d i n g t o B e r n o u l l i ' s t h e o r e m w e g e t t h e p r e s s u r e i n t h e w a k e
Pwake = P0 + ~ (1 - k s) = P0 + ~ 0.8931,
Po b e i n g t h e p r e s s u r e a t z = - o o a n d ~ t h e d e n s i t y of t h e fluid.
T h e t u b e s t u d i e d i n s e c t i o n 2 m i g h t , for i n s t a n c e , b e t h e t u b e in a P i t o t h e a d f o r v e l o c i t y m e a s u r e m e n t , in w h i c h case i t is d e s i r e d t o m e a s u r e t h e s t a g n a t i o n p r e s s u r e . T h e t w o - d i m e n s i o n a l case s t u d i e d in s e c t i o n 3 i n d i c a t e s t h a t , if t h e p r e s s u r e in t h e w a k e is e r r o n e o u s l y t a k e n e q u a l t o t h e s t a g n a t i o n
= P0 + ~ v2, w h e r e v is t h e v e l o c i t y a t z = - o o , t h e v e l o c i t y o b t a i n e d p r e s s u r e
will b e a b o u t 5 p e r c e n t t o o low. I n t h e t h r e e - d i m e n s i o n a l case i t is t h e r e f o r e t o b e e x p e c t e d t h a t t h e r e will b e a n e r r o r of a t l e a s t one or t w o p e r c e n t in t h e v e l o c i t y d e t e r m i n e d .
I n a lecture i n 1950, P r o f e s s o r A l ~ E BEURLING, U p p s a l a , t r e a t e d s o m e h y d r o d y n a m i c p r o b l e m s c o n n e c t e d w i t h wakes, a n d it w a s a t t h a t l e c t u r e t h e a u t h o r g o t t h e i d e a f o r t h i s s t u d y .
R E F E R E N C E
1. GARRETT BIRKHOFF, H y d r o d y n a m i c s , a S t u d y in Logic, F a c t a n d Similitude. Prince- t o n U n i v e r s i t y Press 1950.
394
Tryckt den 15 mars 1957
Uppsala 1957. Almqvist & Wikse]ls Boktryckeri AB
